J is divisble by 12 and 10 - Is J divisible by 24?

Answer says Cannot Be Determined since the prime factors of J are:
2 x 2 x 3 x 5

However, wouldn't the prime factors be: 2 x 2 x 2 x 3 x 5?
12 - 2 x 2 x 3
10 - 2 x 5

From there you could get 24 - 2 x 2 x 2 x 3.

It's been a while since I've worked with prime numbers and factors, so any clarification would be great.

Thanks!

You have to factor out the overlap - you can't just combine all of the prime factors.

Couple of ways to think about this:

First, try some numbers. If J is divisible by 12, what are some possible values for J? 12, 24, 36, 48, 60, 72, ...
If J is divisible by 10, what are some possible values for J? 10, 20, 30, 40, 50, 60, 72, ...

Now, if I'm going to use both statements, I can only take the overlapping possibilities: 60, 120, 180, etc.

Is 60 divisible by 24? No.
Is 120 divisible by 24? Yes.
So the info isn't sufficient.

Now, notice that my smallest possibility for J when looking at both statements was 60. What are the prime factors of 60? 2, 2, 3, 5. Hmm... that's what the answer said, too! How to understand what's going on theoretically so I don't have to list out a bunch of numbers?

Picture a small, one-room building. This building has two windows in it, on different sides of the building. You can't see in either window. Susie goes up to Window 1, looks in, and comes over to tell you that there are a 2, a 2, and a 3 inside that room. (2, 2, and 3 are the prime factors of 12).

So what do you know? There are a 2, a 2, and a 3 in that room.

Then Amy goes up to Window 2, looks in, and comes over to tell you that there are a 2 and a 5 inside that room (2 and 5 are the prime factors of 10).

So what do you know? There are a 2 and a 5 in that room.

Now think about what you definitely know based on the info from BOTH Susie and Amy. There's definitely a 3 in the room. There's also a 5. What about the 2's? Are there definitely three 2's in there? Or could Susie and Amy have been looking at the same 2? We know there are at least two 2's in the room, because Susie reported two separate 2's. But that one 2 that Amy saw - that could have been one of the 2's that Susie saw. So I don't know for sure that there is a 3rd 2 in the room - there might be, but I just don't know.

What's that amount to? 2, 2, 3, 5 - just like we figured out from trying numbers. In other words, each statement gives us true BUT potentially overlapping information - and we have to strip out the overlap when we combine the statements.

Thanks for the explanation, Stacey; I had the exact same question. As a follow-up, I didn't see the concept you explained spelled out in the guide anywhere (unless I just missed it). I've seen this concept arise in a few other questions - is this something that should be intuitive? If not, is there a general rule with respect to this type of analysis?

Thanks.

actually there is: the LEAST COMMON MULTIPLE.

if a number is divisible by both X and Y, then the number must be divisible by the least common multiple of X and Y. ...and that's the biggest number that it MUST be divisible by, although it certainly could be divisible by bigger numbers.

so, in the problem at hand, the number is divisible by 12 and 10; the least common multiple of 12 and 10 is 60, so we know that the number must be divisible by 60.

this doesn't tell us whether the number is divisible by 24, though; as stacey pointed out, the first two multiples of 60 (namely, 60 and 120) give a 'no' and a 'yes' respectively.